= Solution
Orthogonality of the <number states> makes the <coherent state> norm an exponential series:
$$
\langle\alpha|\alpha\rangle=e^{-|\alpha|^2}\sum_{n=0}^\infty\frac{|\alpha|^{2n}}{n!}=1.
$$
Using $\hat a|n\rangle=\sqrt n|n-1\rangle$ and reindexing $m=n-1$,
$$
\hat a|\alpha\rangle=e^{-|\alpha|^2/2}\sum_{n=1}^\infty\frac{\alpha^n}{\sqrt{(n-1)!}}|n-1\rangle
=\boxed{\alpha|\alpha\rangle}.
$$
It is not an <eigenstate> of the <creation operator>. Indeed its supposed eigenvalue would have to be $\alpha^*$, from its expectation, whereas
$$
\left\|(\hat a^\dagger-\alpha^*)|\alpha\rangle\right\|^2
=\langle\hat a\hat a^\dagger\rangle-|\alpha|^2=1.
$$
This nonzero residual holds even at $\alpha=0$.
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